Hollywood heart-throb Johnny Depp has been given an ultimatum by Australia: send your pet dogs back home to the United States or face having them put down.

Depp is in hot water with the Department of Agriculture after failing to declare his two Yorkshire terriers when he flew into Australia on his private jet last month for the filming of the fifth instalment of the Pirates of the Caribbean movie franchise.

“He has decided to bring to our nation two dogs without actually getting proper certification and the proper permits required. Basically, it looks like he snuck them in,” agriculture minister Barnaby Joyce told reporters on Thursday. “We found out he snuck them in because we saw him taking them to a poodle groomer.”

“The reason you can walk through a park in Brisbane and not sort of have in the back of your mind – what happens if a rabid dog comes out and bites me or bites my kid – is because we’ve kept that disease out.

“I’ll tell you how close it is, it’s in Bali, it’s just next door, so this is not fanciful stuff and therefore we’re very diligent about what comes into our nation.”

Within hours, Depp’s Gold Coast residence was surrounded by media waiting for a sight of the dogs.

The author of the petition, 27-year old Sydney woman Namita Sopal, told Guardian Australia she hoped it would start an online movement to put pressure on the government to withdraw its “cruel” threat.

“I love dogs and I thought it was crazy, so that was my motivation,” she said.

The United States embassy in Canberra released a statement warning citizens of Australia’s strict customs and quarantine regulations and advising potential travellers with specific concerns to contact the Australian embassy in Washington DC for more details.

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Lianne Kent, the owner of Happy Dogz, the grooming service where Depp and his wife Amber Heard had taken Pistol and Boo at the weekend, posted on the business’s Facebook page that it had been “an honour” to attend to the pets.

“Their hair was really long and they needed a trim. She [the celebrities’ handler] desperately wanted them trimmed back and their faces styled,” she told the Gold Coast Bulletin. “They wanted the dogs groomed somewhere that’s a bit more private and we’re right near the set anyway.”

Kent did not return Guardian Australia’s calls.

Joyce acknowledged that his tough stance on the issue would come with consequences.

“After that, I don’t expect to be invited to the opening of Pirates of the Caribbean,” he said.